Solar Impulsive Hard X-Ray Emission and Two-Stage Electron Acceleration

Heating and acceleration of electrons in solar impulsive hard X-ray (HXR) flares are studied according to the two-stage acceleration model developed by Zhang for solar 3He-rich events. It is shown that electrostatic H-cyclotron waves can be excited at a parallel phase velocity less than about the electron thermal velocity and thus can significantly heat the electrons (up to 40 MK) through Landau resonance. The preheated electrons with velocities above a threshold are further accelerated to high energies in the flare-acceleration process. The flare-produced electron spectrum is obtained and shown to be thermal at low energies and power law at high energies. In the non-thermal energy range, the spectrum can be double power law if the spectral power index is energy dependent or related. The electron energy spectrum obtained by this study agrees quantitatively with the result derived from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) HXR observations in the flare of 2002 July 23. The total flux and energy flux of electrons accelerated in the solar flare also agree with the measurements.     

Field line topology in the dayside cusp region inferred from low altitude particle observations

Dayside low altitude satellite observations of the pitch angle and energy distribution of electrons and protons in the energy range 1 eV to 100 eV during quite geomagnetic conditions reveal that at times there is a clear latitudinal separation between the precipitating low energy (keV) electrons and protons, with the protons precipitating poleward of the electrons. The high energy (100 keV) proton precipitation overlaps both the low energy (keV) electron and proton precipitation. These observations are consistent with a model where magnetosheath particles stream in along the cusp field lines and are at the same time convected poleward by an electric field.The electrons with energies of a few keV move fast and give the “ionospheric footprint” of the distant cusp. The protons are partly convected poleward of the cusp and into the polar cap. Here the mirroring protons populate the plasma mantle. Equatorward of the cusp the pitch angle distribution of both electrons and protons with energies above a few keV is pancake shaped indicating closed geomagnetic field lines. The 1 keV electrons, penetrate, however, into this region of closed field line structure maintaining an isotropic pitch angle distribution. The intensity is, however, reduced with respect to what it was in the cusp region. It is suggested that these electrons, the lowest energies measured on the satellite, are associated with the entry layer.     

Electron Acceleration and Transport During the November 5, 1998 Solar Flare At ∼13:34 UT

This paper deals with a detailed analysis of spectral and imaging observations of the November 5, 1998 (Hα 1B, GOES M1.5) flare obtained over a large spectral range, i.e., from hard X-rays to radiometric wavelengths. These observations allowed us to probe electron acceleration and transport over a large range of altitudes that is to say within small-scale (a few 10³ km) and large-scale (a few 10⁵ km) magnetic structures. The observations combined with potential and linear force-free magnetic field extrapolations allow us to show that: (i) Flare energy release and electron acceleration are basically driven by loop–loop interactions at two independent, low lying, null points of the active region magnetic field; (ii) <300 keV hard X-ray-producing electrons are accelerated by a different process (probably DC field acceleration) than relativistic electrons that radiate the microwave emission; and (iii) although there is evidence that hard X-ray and decimetric/metric radio-emitting electrons are produced by the same accelerator, the present observations and analysis did not allow us to find a clear and direct magnetic connection between the hard X-ray emitting region and the radio-emitting sources in the middle corona.     

Comparisons of solar flare X-ray producing and escaping electrons from ∼ 2 TO 100 keV

Using observations from the ISEE-3 spacecraft, we compare the X-ray producing electrons and escaping electrons from a solar flare on 8 November, 1978. The instantaneous 5 to 75 keV electron spectrum in the X-ray producing region is computed from the observed bremsstrahlung X-ray spectrum. Assuming that energy loss by Coulomb collisions (thick target) is the dominant electron loss process, the accelerated electron spectrum is obtained. The energy spectrum of the escaping electrons observed from 2 to 100 keV differs significantly from the spectra of the X-ray producing electrons and of the accelerated electrons, even when the energy loss which the escaping electrons experienced during their travel from the Sun to the Earth is taken into account. The observations are consistent with a model where the escaping electrons come from an extended X-ray producing region which ranges from the chromosphere to high in the corona. In this model the low energy escaping electrons (2–10 keV) come from the higher part of the extended X-ray source where the overlying column density is low, while the high energy electrons (20–100 keV) come from the entire X-ray source.     

The aries auroral modelling campaign: characterization and modelling of an evening auroral arc observed from a rocket and a ground-based line of meridian scanners

An auroral arc system excited by soft electrons was studied with a combination of in situ rocket measurements and optical tomographic techniques, using data from a photometer on a horizontal, spinning rocket and a line of three meridian scanning photometers. The ground-based scanner data at 4709, 5577, 8446 and 6300 Å were successfully inverted to provide a set of volume emission rate distributions in the plane of the rocket trajectory, with a basic time resolution of 24 s. Volume emission rate profiles, derived from these distributions peaked at about 150 km for 5577 and 4709 Å, while the 8446 Å emission peaked at about 170 km with a more extended height distribution. The rocket photometer gave comparable volume emission rate distributions for the 3914 Å emission as reported in a separate paper by McDade et al. (1991, Planet. Space Sci. 39, 895). Instruments on the rocket measured the primary electron flux during the flight and, in particular, the flux precipitating into the auroral arc overflown at apogee (McEwen et al., 1991; in preparation). The local electron density and temperature were measured by probes on the rocket (Margot and McNamara (1991; Can. J. Phys. 69, 950). The electron density measurements on the downleg were modelled using ion production rate data derived from the optical results. Model calculations of the emission height profile based on the measured electron flux agree with the observed profiles. The height distribution of the N₂⁺ emission in the equatorward band, through which the rocket passed during the descent, was measured by both the rocket and the ground-based tomographic techniques and the results are in good agreement. Comparison of these profiles with model profiles indicates that the exciting primary spectrum may be represented by an accelerated Maxwellian or a Gaussian distribution centered at about 3 keV. This distribution is close to what would be obtained if the electron flux exciting the poleward form were accelerated by a 1–2 kV upward potential drop. The relative height profiles for the volume emission rate of the 5577 Å OI emission and the 4709 Å N₂⁺ emission were almost indistinguishable from each other for both the forms measured, with ratios in the range 38–50; this is equivalent to I(5577)/I(4278) ratios of 8–10. The auroral intensities and intensity ratios measured in the magnetic zenith from the ground during the period before and during the rocket flight are consistent with the primary electron fluxes and height distributions measured from the rocket. Values of I(5577)/I(4278) in the range 8–10 were also measured directly by the zenith ground photometers over which the arc system passed. These values are slightly higher than those reported by Gattinger and Vallance-Jones (1972) and this may possibly indicate an enhancement of the atomic oxygen concentration at the time of the flight. Such an enhancement would be consistent with our result, that the observed values of I(5577) and I(8446) are also significantly higher than those modelled on the basis of the electron flux spectrum measured at apogee.     

Rocket observation of conjugate photoelectrons in the predawn ionosphere

Photoelectron flux in the energy range 6–70 eV coming from the sunlight conjugate ionosphere has been measured directly by the rocket borne low energy electron spectrometer in the altitude region of 210–350 km. Pitch angle distribution of the measured flux is nearly isotropic, the flux decreasing slightly with pitch angle. The photoelectron fluxes measured at 350 km at the energies of 15 and 30 eV are 3 × 10⁶ and 1 × 10⁶ (cm² s str eV)^?1 respectively which decrease to 1 × 10⁶ and 1 × 10⁵ at 250 km at the same energies. These values are consistent with the vertical profile of the 630 nm airglow intensity measured simultaneously. The fluxes obtained near apogee show peaks in the range 20–30 eV which also appear in the daytime photoelectron flux, indicating reduced loss of electrons during the passage from the conjugate ionosphere through the plasmasphere at the low geomagnetic latitude where observation was made. Photoelectron fluxes observed below the apogee height are compared to the calculated fluxes to investigate the interaction of electrons with the atmospheric species during the passage in the ionosphere. Calculated fluxes obtained by using continuous slowing-down approximation and neglecting pitch angle scattering are in good agreement with the observations although there still remain disagreements in detailed comparison which may be ascribed to the assumptions inherent in the calculation and/or to the uncertainties of the input data for the calculation.     

Observations of soft electron flux during SAR arc event

Attempts to test the validity of the soft electron flux hypothesis for the excitation of SAR arcs have heretofore met with no apparent success. However, observations of other emissions during the times of some SAR arcs repeatedly favour the presence of the soft electron flux in the topside ionosphere.Although no new cases of coincident measurements of SAR arc intensities and the corresponding soft electron flux values are available, an instance has been analyzed in which an SAR arc was examined shortly after an OGO 6 measurement of the soft electron flux. The SAR arc observations were then interrupted by the weather, but in view of the behaviour of the electron flux during the arc development it is believed that this event lends strong observational support to the hypothesis that soft electrons can be an excitation source for SAR arcs.In the case considered, the precipitation of soft electron flux peaks at about the same location and time of occurrence as the SAR arc. The peak value is found to be 5.2 × 10⁸ cm^?2 sec^?1, which is more than adequate to excite the observed arc.     

Energetics of a double flare on November 8, 1980

Here we complete an energy balance analysis of a double impulsive hard X-ray flare. From spatial observations, we deduce both flares probably occur in the same loop within the resolution of the data. For the first flare, the energy in the fast electrons (assuming a thick-target model) is comparable to the convective up-flow energy, suggesting that these are related successive modes of energy storage and transfer. The total energy lost through radiation and conduction, 2.0 × 10²⁸ erg, is comparable to the energy in fast electrons 2.5 × 10²⁸ erg. For the second flare, the energy in the fast electrons is more than one order of magnitude greater than the energy of the convective up-flow. Total energy losses are within a factor of two lower than the calculated fast electron energy. We interpret the observations as showing that the first flare occurred in a small loop with fast electrons heating the chromosphere and resulting in chromospheric evaporation increasing the density in the loop. For the second flare most of the heating occurred at the electron acceleration site. The two symmetrical components of the Ca xix resonance line and a high velocity down-flow of 115 km s ^–1 observed at the end of the second hard X-ray burst are consistent with the flare eruption (reconnection) region being high in the flare loop. The estimated altitude of the acceleration site is 5500 km above the photosphere.     

Time-dependent studies of the aurora: Effects of particle precipitation on the dynamic morphology of ionospheric and atmospheric properties

A self-consistent, time-dependent numerical model of the aurora and high-latitude ionos-phere has been developed. It is used to study the response of ionospheric and atmospheric properties in regions subjected to electron bombardment. The time history of precipitation events is arbitrarily specified and computations are made for a variety of electron spectral energy distributions and flux magnitudes. These include soft electron precipitation, such as might occur on the poleward edge of the auroral oval and within the magnetospheric cleft, and harder spectra representative of particle precipitation commonly observed within and on the equatorward edge of the auroral oval. Both daytime and night-time aurorae are considered. The results of the calculations show that the response of various ionospheric and atmospheric parameters depends upon the spectral energy distribution and flux magnitudes of the precipitating electrons during the auroral event. Various properties respond with different time constants that are influenced by coupling processes described by the interactive model. The soft spectrum aurora affects mainly the ionospheric F region, where it causes increases in the electron density, electron temperature and the 6300 Å red line intensity from normal quiet background levels during both daytime and night-time aurora. The fractional variation is greater for the night-time aurora. The hard spectrum aurorae, in general, do not greatly affect the F-2 region of the ionosphere; however, in the F-1 and E regions, large increases from background conditions are shown to occur in the electron and ion temperatures, electron and ion densities, airglow emission rates and minor neutral constituent densities during the build-up phase of the auroral event. During the decay phase of the aurora, most of these properties decrease at nearly the same rate as the specified particle precipitation flux. However, some ionospheric and atmospheric species have a long memory of the auroral event. The odd nitrogen species N(⁴S) and NO probably do not ever reach steady-state densities between auroral storms.     

Pitch angle and energy distributions of auroral electrons measured by the ESRO 4 satellite

Energy spectra and angular distributions of auroral electrons in the energy range 0.2–16 keV measured by the low-altitude polar orbiting satellite ESRO 4 are presented. The observations were made in the altitude range 800–1000 km near magnetic midnight. Energy-time spectrograms show inverted-V structures with peaked energy spectra. The inverted-V events are associated with anisotropic electron pitch angle distributions peaked at 0 deg. Frequently these distributions have a maximum also at 90 deg. Measurements of >43 keV electrons indicate that the acceleration probably occurs on closed field lines. It is found that many properties of the observed particle distributions can be explained by acceleration in an electric field parallel to the magnetic field lines, if trapping of particles under an increasing potential drop is included in the model.     

Electron precipitation observations from a rocket flight through the dayside auroral oval

Energy spectra of electrons between 30 eV and 18 keV were obtained with a spectrometer on a Black Brant rocket launched from Cape Parry, N.W.T. (Λ = 75.2°) on December 6, 1974 to study the dayside magnetospheric cleft. The rocket flew to an apogee of 236 km and travelled poleward to 80° invariant latitude. The cleft was observed to extend from 76.9 to 78.4° invariant latitude. Equatorward of this electrons of a few keV energy were observed with a total energy flux of up to 2 erg/cm² sec ster. Variable fluxes of electrons with a spectrum fitted by a Maxwellian distribution of 150 eV characteristic energy were observed through most of the cleft. One inverted V structure was crossed. In that region, the electron energy increased to 650 eV and a total energy flux of 8 erg/cm² sec ster was measured. The event was a temporal one and only a few km in width, as deduced from optical data. Fluxes of about 10⁻² erg/cm² sec ster were recorded poleward of the cleft.     

On the energy dependence of the ring current proton precipitation

Low altitude satellite measurements of protons in the 1–100 keV range indicate two energy dependent proton precipitation boundaries. At low invariant latitudes mostly below 60° there is a region of moderately weak proton precipitation. The poleward boundary of this region tends to be at higher latitudes for the high energy protons than for the low energy protons. At high invariant latitudes there is a region where both the low and high energy protons precipitate with an isotropic pitch-angle distribution. The equatorward boundary of this region tends to be at lower latitudes for protons with energy more than 100 keV than for those in the 1–6 keV range. This region with isotropic pitch-angle distribution is located well outside the plasmapause both for the 1–6 and 100-keV protons.Between these two precipitation zones there is a region where the proton pitch-angle distribution is highly anisotropic with almost no protons in the loss cone. This region tends to be wider and more pronounced in the 1–6 than in the 100-keV protons.These findings lend further support to the mechanism of ion-cyclotron instability as the cause of proton pitch-angle diffusion in the low and intermediate regions. The process responsible for the strong diffusion at auroral latitudes has not yet been identified.     

Impulsive bursts of relativistic electrons discovered during Ulysses' traversal of Jupiter's dusk-side magnetosphere

During its flyby of Jupiter in February 1992, the Ulysses spacecraft passed through the Southern Hemisphere dusk-side Jovian magnetosphere, a region not previously explored by spacecraft. Among the new findings in this region were numerous, sometimes periodic, bursts of high energy electrons with energies extending from less than 1.5 MeV to beyond 16 MeV. These bursts were discovered by the High Energy Telescope (HET) and the Kiel Electron Telescope (KET) of the COSPIN Consortium. In this paper we provide a detailed analysis of observations related to the bursts using HET measurements. At the onset of bursts, the intensity of > 16 MeV electrons often rose by a factor of > 100 within 1 min, and multiple, pulsed injections were sometimes observed. The electron energy spectrum also hardened significantly at the onset of a burst. In most bursts anisotropy measurements indicated initial strong outward streaming of electrons along magnetic field lines that connect to the southern polar regions of Jupiter, suggesting that the acceleration and/or injection region for the electrons lies at low altitudes near the South Pole. The initial strong outward anisotropies relaxed to strong field-aligned bidirectional anisotropies later in the events. The bursts sometimes appeared as isolated events, but at other times appeared in quasi-periodic series with a period of 40 min. For smaller events shorter periods of the order 2–3 min were also observed in a few cases. For large events, multiple injections were sometimes observed in the first few minutes of the event. Radio bursts identified by the Ulysses URAP experiment in the frequency range 1–50 kHz were correlated with many of the electron bursts, and comparison of the time-intensity profiles for radio and electrons shows that the radio emission typically started several minutes before the electron intensity increase was observed. For the strongest electron bursts, small increases in the low energy (> 0.3 MeV) proton counting rates were also observed. Using a computerized identification algorithm to pick out bursts from the data record using a consistent set of criteria, 121 events were identified as electron bursts during the outbound pass, compared to only three events that satisfied the same criteria during the inbound pass through the day-side magnetosphere. No similar electron burst events have been found outside the magnetopause. Estimates of the electron content of a typical large burst (> 10²⁷ electrons) suggest that these bursts may make significant contributions to the fluxes of electrons observed in Jupiter's outer magnetosphere, and in interplanetary space.     

The formation of high energy electron fluxes in the inner radiation belt of earth

Measurements on board the low altitude polar orbiting satellite Intercosmos-17 /nearly circular orbit h = 500 km, i = 83.5°/ have shown relatively high fluxes of high energy electrons /E_e > 100 MeV and E_e > 300 MeV respeetively/ at minimum-B-equator. Computation of the electron production spectra assuming the interaction of high energy protons of the inner radiation belt with residual atmosphere is made. The considered mechanism can explain the enhanced flux of high energy electrons registered in the Brazil magnetic anomaly.     

Proton observations supporting the ion cyclotron wave heating theory of SAR arc formation

Low altitude satellite observations of precipitated and locally mirroring protons during periods of ground-based SAR arc observations are presented. The SAR arcs are found to be located in a region with significantly enhanced proton pitch angle scattering and enhanced electron temperature, but inside the plasmapause where the proton pitch angle distribution is anisotropic. The increase in the pitch angle scattering takes place in a localized region having a width of a few tenths of a L-value. The observations can favourably be accounted for by the Cornwall et al. (1971) theory for the SAR arc formation. Using observed proton fluxes and typical energy spectra, the expected Hβ intensity in the SAR arc region is estimated to be a few Rayleighs, and the energy flux from precipitated protons above a few keV to be 10^?2?10^?1erg/cm²s. These estimates are in reasonable agreement with previously published theoretical and experimental values. Simultaneous groundbased observations of Hα emissions were found in the region of intense, isotropic proton precipitation located outside the plasmapause.     

Topside optical view of the dayside cleft aurora

Photometers on the ISIS-II spacecraft provide a view of the atomic oxygen 5577 and 6300 Å emissions and the $\text{N}{}_2{}^{\mathrm{+}}\text{3914}\text{A}\text{?}$ emission detected as dayside aurora in the magnetospheric cleft region. The 6300 Å emission forms a continuous and permanent band across the noon sector, at about 78° invariant latitude, with a defined region of maximum intensity that is never less than 2kR (uncorrected for albedo), and is centred near magnetic noon. There are significant differences in the intensity patterns on either side of noon and their responses to geomagnetic activity. Discrete 3914 Å auroral forms appear within this region, at preferred locations that cannot be precisely specified, but which tend to the poleward edge of the 6300 Å emission in the evening, and the equatorward edge in the morning where the difference between the two emissions is greatest. It is concluded that the discrete auroras observed by all-sky cameras in the day sector do follow the 6300 Å emission through the cleft region, though a definite cleft boundary is not defined. Substantial 6300 Å emission having a peak intensity near noon is also seen in the low latitude “outer auroral belt”, while the diffuse 3914 Å emission tends to show a relative minimum near noon. On the morning side the 3914 Å intensity is displaced to lower latitude and earlier local times, compared to the 6300 Å emission.     

Precipitation of > 115 kev protons in the evening and forenoon sectors in relation to the magnetic activity

Data from a low altitude polar orbiting satellite, on auroral protons >115 keV in the evening and forenoon sectors, are presented.In the forenoon sector there is a weak but fairly steady precipitation at Λ ≈ 75° during quiet conditions. This precipitation is situated at higher invariant latitudes at local noon than at local dawn and can probably be ascribed to the high energy tail of the polar cleft protons. During moderately disturbed conditions, especially during the recovery phase of geomagnetic storms, there are some seemingly more “impulsive” precipitation events at Λ ≈ 65°. During very disturbed conditions these two precipitation zones in the forenoon sector seem to merge.In the evening sector a rather sharp equatorward boundary of the main precipitation, at Λ ≈ 69° during quiet conditions, varies fairly smoothly from pass to pass. South of this boundary, at invariant latitudes around 62°, there is a steady weak drizzle from the radiation belt. Due to a longitudinal effect this drizzle, as recorded by the satellite, shows a diurnal variation.The equatorward boundaries of the main precipitation at both local times move equatorward with increasing ring current strength. When D_st gets less than about — 100nT, the poleward boundaries are found to move equatorward too. From an attempt to reveal some of the substorm-dependent changes of the precipitation it is found that an equatorward shift of the precipitation areas takes place during, or just prior to, the substorm expansive phase, accompanied by a large intensity increase in the evening sector, whereas the recovery phase is linked with a poleward expansion of the precipitation at both local times.     

A study of the dynamics of a discrete auroral arc

High resolution electric field and particle data, obtained by the S23L1 rocket crossing over a discrete prebreakup arc in January 1979, are studied in coordination with ground observations (Scandinavian Magnetometer Array—SMA, TV and all-sky cameras) in order to clarify the electrodynamics of the arc and its surroundings. Height-integrated conductivities have been calculated from the particle data, including the ionization effects of precipitating protons and assuming a steady state balance between ion production and recombination losses. High resolution optical information of arc location relative to the rocket permitted a check of the validity of this assumption for each flux tube passed by the rocket. Another check was provided by a comparison between calculated (equilibrium values) and observed electron densities along the rocket trajectory. A way to compensate for the finite precipitation time when calculating the electron densities is outlined. The height-integrated HalI-Pedersen conductivity ratio is typically 1.4 within the arc and about 1 at the arc edges, indicative of a relatively softer energy spectrum there. The height-integrated conductivities combined with the DC electric field measurements permitted calculation of the horizontal ionospheric current vectors (J_⊥), Birkeland currents (from div J_⊥) and energy dissipation through Joule heating (Σ_pE²). An eastward current of typically 1 A m^?1 was found to be concentrated mainly to the arc region and equatorward of it. A comparison has been made with the equivalent current system deduced from ground based magnetometer data (SMA) showing a generally good agreement with the rocket results. An intense Pedersen current peak (1.2 A m^?1) was found at the southern arc edge. This edge constituted a division line between a very intense (> 10 μA m^?1) and localized (~ 6 km) downward current sheet to the south, probably carried by upward flowing cold ionospheric electrons and a more extended upward current sheet (> 10 μA m^?2) over the arc carried by measured precipitating electrons. Joule and particle heating across the arc were anticorrelated, consistent with the findings of Evans et al. (1977) with a total value of about 100mW m^?2.     

Nearly simultaneous observations of the conjugate polar cusp regions

The first simultaneous (within 6 min) observations of the low altitude polar cusp regions in the conjugate hemispheres are reported here based on two events detected by the DMSP-F2 and F4 satellites within the same geomagnetic local time sector. It is found that the electron spectra in the cusp are identical in the opposing hemispheres. In one case the observed latitudinal location and extent of the cusps are the same at the two hemispheres. However, in the other case the location of the equatorward boundary of the cusp regions differs by about 2° with drastically different spatial features. It is also found that in one of the events the plasma sheet electron precipitation regions overlap with the cusp regions at lower latitude in both hemispheres. The poleward boundary of these overlapping regions is located at the same latitude on either hemisphere, suggesting that this is the latitude of the last closed field line and that the cusp electrons are present on both closed and open magnetic field lines.     

The relationship of electron plasma oscillations to type III radio emissions and low-energy solar electrons

An extensive study of the IMP-6 and IMP-8 plasma and radio wave data has been performed to try to find electron plasma oscillations associated with type III radio noise bursts and low energy solar electrons. This study shows that electron plasma oscillations are seldom observed in association with solar electron events and type III radio bursts at 1.0 AU. In nearly four years of observations only one event was found in which electron plasma oscillations are clearly associated with solar electrons. Numerous cases were found in which no electron plasma oscillations with field strengths greater than 1 V/m could be detected even though electrons from the solar flare were clearly detected at the spacecraft.For the one case in which electron plasma oscillations are definitely produced by the electrons ejected by the solar flare, the electric field strength is very small, only about 100 V/m. This field strength is about a factor of ten smaller than the amplitude of electron plasma oscillations generated by electrons streaming into the solar wind from the bow shock. Electromagnetic radiation, believed to be similar to the type III radio emission, is also observed coming from the region of more intense electron plasma oscillations upstream of the bow shock. Quantitative calculations of the rate of conversion of the plasma oscillation energy to electromagnetic radiation are presented for plasma oscillations excited by both solar electrons and electrons from the bow shock. These calculations show that neither the type III radio emissions nor the radiation from upstream of the bow shock can be adequately explained by a current theory for the coupling of electron plasma oscillations to electromagnetic radiation. Possible ways of resolving these difficulties are discussed.